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√4.1, no calculator. You know √4 = 2, so how far past 2 is it?
Zoom in on the graph of √x at the point 4, 2 and keep zooming. The curve flattens until it's indistinguishable from its tangent line. At 1000× zoom, the widest gap between them on screen is 0.04 pixels. Close enough to a point, every smooth curve is a straight line, so the tangent line is a free estimate.
The tangent at x = 4 has slope 1/2√4 = 1/4. Step 0.1 along it and you rise 0.1 × 1/4 = 0.025. So √4.1 ≈ 2.025. The true value is 2.02485…, off by 0.00015, right to three decimal places.
Calculus #23 of 113. Full playlist on my channel.
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2026/10/06
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